Download - Research Methods William G. Zikmund, Ch23
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Business Research Methods
William G. Zikmund
Chapter 23
Bivariate Analysis: Measures of Associations
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Measures of Association
• A general term that refers to a number of bivariate statistical techniques used to measure the strength of a relationship between two variables.
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Relationships Among Variables
• Correlation analysis
• Bivariate regression analysis
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Type ofMeasurement
Measure of Association
Interval andRatio Scales
Correlation CoefficientBivariate Regression
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Type ofMeasurement
Measure of Association
Ordinal Scales Chi-squareRank Correlation
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Type ofMeasurement
Measure of Association
NominalChi-Square
Phi CoefficientContingency Coefficient
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Correlation Coefficient• A statistical measure of the covariation or
association between two variables.
• Are dollar sales associated with advertising dollar expenditures?
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The Correlation coefficient for two variables, X and Y is
xyr.
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Correlation Coefficient
• r
• r ranges from +1 to -1
• r = +1 a perfect positive linear relationship
• r = -1 a perfect negative linear relationship
• r = 0 indicates no correlation
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22YYiXXi
YYXXrr ii
yxxy
Simple Correlation Coefficient
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22yx
xyyxxy rr
Simple Correlation Coefficient
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= Variance of X
= Variance of Y
= Covariance of X and Y
2x2y
xy
Simple Correlation Coefficient Alternative Method
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X
Y
NO CORRELATION
.
Correlation Patterns
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X
Y
PERFECT NEGATIVECORRELATION - r= -1.0
.
Correlation Patterns
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X
Y
A HIGH POSITIVE CORRELATIONr = +.98
.
Correlation Patterns
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Pg 629
589.5837.17
3389.6r
712.99
3389.6 635.
Calculation of r
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Coefficient of Determination
Variance
variance2
Total
Explainedr
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Correlation Does Not Mean Causation
• High correlation
• Rooster’s crow and the rising of the sun– Rooster does not cause the sun to rise.
• Teachers’ salaries and the consumption of liquor – Covary because they are both influenced by a
third variable
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Correlation Matrix
• The standard form for reporting correlational results.
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Correlation Matrix
Var1 Var2 Var3
Var1 1.0 0.45 0.31
Var2 0.45 1.0 0.10
Var3 0.31 0.10 1.0
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Walkup’s First Laws of Statistics
• Law No. 1– Everything correlates with everything, especially
when the same individual defines the variables to be correlated.
• Law No. 2– It won’t help very much to find a good correlation
between the variable you are interested in and some other variable that you don’t understand any better.
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• Law No. 3– Unless you can think of a logical reason why
two variables should be connected as cause and effect, it doesn’t help much to find a correlation between them. In Columbus, Ohio, the mean monthly rainfall correlates very nicely with the number of letters in the names of the months!
Walkup’s First Laws of Statistics
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Going back to previous conditionsGoing back to previous conditions Tall men’s sonsTall men’s sons
DICTIONARYDICTIONARYDEFINITIONDEFINITION
GOING OR GOING OR MOVINGMOVINGBACKWARDBACKWARD
Regression
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Bivariate Regression
• A measure of linear association that investigates a straight line relationship
• Useful in forecasting
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Bivariate Linear Regression
• A measure of linear association that investigates a straight-line relationship
• Y = a + bX
• where
• Y is the dependent variable
• X is the independent variable
• a and b are two constants to be estimated
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Y intercept
• a
• An intercepted segment of a line
• The point at which a regression line intercepts the Y-axis
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Slope
• b
• The inclination of a regression line as compared to a base line
• Rise over run
• D - notation for “a change in”
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Y
160
150
140
130
120
110
100
90
80
70 80 90 100 110 120 130 140 150 160 170 180 190
X
My lineYour line
.
Scatter Diagram and Eyeball Forecast
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130
120
110
100
90
80
80 90 100 110 120 130 140 150 160 170 180 190X
Y
.
XaY ˆˆ
XY
Regression Line and Slope
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X
Y
160
150
140
130
120
110
100
90
80
70 80 90 100 110 120 130 140 150 160 170 180 190
Y “hat” forDealer 3
Actual Y forDealer 7
Y “hat” for Dealer 7
Actual Y forDealer 3
Least-Squares Regression Line
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130
120
110
100
90
80
80 90 100 110 120 130 140 150 160 170 180 190X
Y
}}
{Deviation not explained
Total deviation
Deviation explained by the regression
Y
.
Scatter Diagram of Explained and Unexplained Variation
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The Least-Square Method
• Uses the criterion of attempting to make the least amount of total error in prediction of Y from X. More technically, the procedure used in the least-squares method generates a straight line that minimizes the sum of squared deviations of the actual values from this predicted regression line.
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The Least-Square Method
• A relatively simple mathematical technique that ensures that the straight line will most closely represent the relationship between X and Y.
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Regression - Least-Square Method
n
iie
1
2 minimumis
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= - (The “residual”)
= actual value of the dependent variable
= estimated value of the dependent variable (Y hat)
n = number of observations
i = number of the observation
ie iY iY
iY
iY
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The Logic behind the Least-Squares Technique
• No straight line can completely represent every dot in the scatter diagram
• There will be a discrepancy between most of the actual scores (each dot) and the predicted score
• Uses the criterion of attempting to make the least amount of total error in prediction of Y from X
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XYa ˆ
Bivariate Regression
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22
ˆ
XXn
YXXYn
Bivariate Regression
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= estimated slope of the line (the “regression coefficient”)
= estimated intercept of the y axis
= dependent variable
= mean of the dependent variable
= independent variable
= mean of the independent variable
= number of observations
X
Y
n
a
Y
X
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625,515,3759,24515
875,806,2345,19315ˆ
625,515,3385,686,3
875,806,2175,900,2
760,170
300,93 54638.
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12554638.8.99ˆ a
3.688.99
5.31
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12554638.8.99ˆ a
3.688.99
5.31
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XY 546.5.31ˆ
89546.5.31
6.485.31
1.80
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XY 546.5.31ˆ
89546.5.31
6.485.31
1.80
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165546.5.31ˆ
129) value Y (Actual 7Dealer
7
Y6.121
95546.5.31ˆ
)80 value Y (Actual 3Dealer
3
Y4.83
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99 YYei 5.9697
5.0
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165546.5.31ˆ
129) value Y (Actual 7Dealer
7
Y6.121
95546.5.31ˆ
)80 value Y (Actual 3Dealer
3
Y4.83
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99 YYei 5.9697
5.0
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119546.5.319 Y
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F-Test (Regression)
• A procedure to determine whether there is more variability explained by the regression or unexplained by the regression.
• Analysis of variance summary table
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Total Deviation can be Partitioned into Two Parts
• Total deviation equals
• Deviation explained by the regression plus
• Deviation unexplained by the regression
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“We are always acting on what has just finished happening. It happened at least
1/30th of a second ago.We think we’re in the present, but we aren’t. The present we
know is only a movie of the past.”Tom Wolfe in
The Electric Kool-Aid Acid Test
.
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iiii YYYYYY ˆ ˆ
Partitioning the Variance
Total deviation
=Deviation explained by the regression
Deviation unexplained by the regression (Residual error)
+
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= Mean of the total group
= Value predicted with regression equation
= Actual value
Y
Y
iY
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222 ˆ ˆ iiii YYYYYY
Total variation explained
=Explained variation
Unexplained variation (residual)
+
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SSeSSrSSt
Sum of Squares
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Coefficient of Determination r2
• The proportion of variance in Y that is explained by X (or vice versa)
• A measure obtained by squaring the correlation coefficient; that proportion of the total variance of a variable that is accounted for by knowing the value of another variable
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Coefficient of Determination r2
SStSSe
SStSSr
r 12
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Source of Variation
• Explained by Regression
• Degrees of Freedom– k-1 where k= number of estimated constants
(variables)
• Sum of Squares – SSr
• Mean Squared– SSr/k-1
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Source of Variation
• Unexplained by Regression
• Degrees of Freedom– n-k where n=number of observations
• Sum of Squares – SSe
• Mean Squared– SSe/n-k
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r2 in the Example
875.4.882,3
49.398,32 r
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Multiple Regression
• Extension of Bivariate Regression
• Multidimensional when three or more variables are involved
• Simultaneously investigates the effect of two or more variables on a single dependent variable
• Discussed in Chapter 24
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Correlation Coefficient, r = .75
Correlation: Player Salary and Ticket Price
-20-10
0102030
1995 1996 1997 1998 1999 2000 2001
Change in TicketPrice
Change inPlayer Salary
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